A jsfiddle example would help a lot but my guess is you're using done/fail/always like options:

$.ajax( {
    done: myDoneCallback
} );

But done/fail/always aren't options, they're methods of the returned jqXHR object:

$.ajax().done( myDoneCallback );

The success/error/complete that are deprecated are the *methods* too, not the options.

// Deprecated
$.ajax().success( myDoneCallback );

// Not Deprecated
$.ajax( {
    success: myDoneCallback
} );

Regarding statusCode, the documentation states:

An object of numeric HTTP codes and functions to be called when the response has the corresponding code. For example, the following will alert when the response status is a 404:

>        $.ajax({
>          statusCode: {
>            404: function() {
>              alert("page not found");
>            }
>          }
>        });
>    

If the request is successful, the status code functions take the same parameters as the success callback; if it results in an error (including 3xx redirect), they take the same parameters as the error callback.

So a statusCode callback will always be called regardless of the success/error status of the ajax call. Exact, relative timing, isn't given nor should it (because you shouldn't rely on it) but callbacks are called in the order they are added. For options callbacks, I don't remember which comes first though between complete and statusCode callbacks but success/error will be called prior to anything else.

For exact ordering, use the methods.

// Provided the request is successful
$.ajax().done( callNumber1 ).statusCode( {
    202: callNumber2
} ).always( callNumber3 ).statusCode( {
    202: callNumber4
} ).done( callNumber5 );

#14012 is a duplicate of this ticket.

Thank you, this addresses my issue. If you could include some or all of this information in the documentation, that would be helpful.

The docs seem clear enough, further clarification can be had by asking targeted questions on our forum or StackOverflow. Making the ajax docs longer to clarify obscure points will make it harder to understand.